# Geometry

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Publication date
1987

Topics
mathematics, geometry, analytic geometry, vectors, straight line, mir publishers, mir books, conic sections, quadric surfaces, tangents, curves, differential geometry, history, curvature, torsion, axioms, non-Euclidean, projective, elementary problems, polyhedra, angles, volumes

Collectionopensource

LanguageEnglish

ABOUT THE BOOK

This is a manual for the students of universities and teachers' training colleges. Containing the compulsory course of geometry, its particular impact is on elementary topics. The book is, therefore, aimed at professional training of the school or university teacher-to-be. The first part, analytic geometry, is easy to assimilate, and actually reduced to acquiring skills in applying algebraic methods to elementary geometry.

The second part, differential geometry, contains the basics of the theory of curves and surfaces. The third part, foundations of geometry, is original. The fourth part is devoted to certain topics of elementary

geometry. The book as a whole must interest the reader in school or university teacher's profession.

The book was translated from the Russian by Leonid Levant, Aleksandr Repyev and Oleg Efimov and published by Mir in 1987.

All credits to the original uploader.

Contents

Preface 10

Part One. Analytic Geometry 11

Chapter I. Rectangular Cartesian Coordinates in the Plane 11

1. Introducing Coordinates in the Plane 11

2. Distance Between Two Points 12

3. Dividing a Line Segment in a Given Ratio 13

4. Equation of a Curve. Equation of a Circle 15

5. Parametric Equations of a Curve 17

6. Points of Intersection of Curves 19

7. Relative Position of Two Circles 20

Exercises to Chapter I 21

Chapter II. Vectors in the Plane 26

1. Translation 26

2. Modulus and the Direction of a Vector 28

3. Components of a Vector 30

4. Addition of Vectors 30

5. Multiplication of a Vector by a Number 31

6. Collinear Vectors 32

7. Resolution of a Vector into Two Non-Collinear Vectors 33

8. Scalar Product 34

Exercises to Chapter II 36

Chapter III. Straight Line in the Plane 38

1. Equation of a Straight Line. General Form 38

2. Position of a Straight Line Relative to a Coordinate System 40

3. Parallelism and Perpendicularity Condition for Straight Lines 41

4. Equation of a Pencil of Straight Lines 42

5. Normal Form of the Equation of a Straight Line 43

6. Transformation of Coordinates 44

7. Motions in the Plane 47

8. Inversion 47

Exercises to Chapter III

Chapter IV. Conic Sections 53

1. Polar Coordinates 53

2. Conic Sections 54

3. Equations of Conic Sections in Polar Coordinates 56

4. Canonical Equations of Conic Sections in Rectangular Cartesian Coordinates 57

5. Types of Conic Sections 59

6. Tangent Line to a Conic Section 62

7. Focal Properties of Conic Sections 65

8. Diameters of a Conic Section 67

9. Curves of the Second Degree 69

Exercises to Chapter IV 71

Chapter V. Rectangular Cartesian Coordinates and Vectors in Space 76

1. Cartesian Coordinates in Space. Introduction 76

2. Translation in Space 78

3. Vectors in Space 79

4. Decomposition of a Vector into Three Non-coplanar Vectors 80

5. Vector Product of Vectors 81

6. Scalar Triple Product of Vectors 83

7. Affine Cartesian Coordinates, 84

8. Transformation of Coordinates 85

9. Equations of a Surface and a Curve in Space 87

Exercises to Chapter V 89

Chapter VI.

Plane and a Straight Line in Space 95

1. Equation of a Plane 95

2. Position of a Plane Relative to a Coordinate System 96

3. Normal Form of Equations of the Plane 97

4. Parallelism and Perpendicularity of Planes 98

5. Equations of a Straight Line 99

6. Relative Position of a Straight Line and a Plane, of Two Straight Lines 100

7. Basic Problems en Straight Lines and Planes 102

Exercises to Chapter VI 103

Chapter VII. Quadric Surfaces 109

1. Special System of Coordinates 109

2. Classification of Quadric Surfaces 112

3. Ellipsoid 113

4. Hyperboloids 115

5. Paraboloids 116

6. Cone and Cylinders 118

7. Rectilinear Generators on Quadric Surfaces 119

8. Diameters and Diametral Planes of a Quadric Surface 120

9. Axes of Symmetry for a Curve. Planes of Symmetry for a Surface 122

Exercises to Chapter VII 123

Part Two.

Differential Geometry 126

Chapter VIII. Tangent and Osculating Planes of Curve 126

1. Concept of Curve 126

2. Regular Curve 127

3. Singular Points of a Curve 128

4. Vector Function of Scalar Argument 129

5. Tangent to a Curve 131

6. Equations of Tangents for Various Methods of Specifying a Curve 132

7. Osculating Plane of a Curve 134

8. Envelope of a Family of Plane Curves 136

Exercises to Chapter VIII 137

Chapter IX. Curvature and Torsion of Curve 140

1. Length of a Curve 140

2. Natural Parametrization of a Curve 142

3. Curvature 142

4. Torsion of a Curve 145

5. Frenet Formulas 147

6. Evolute and Evolvent of a Plane Curve 148

Exercises to Chapter IX 149

Chapter X. Tangent Plane and Osculating Paraboloid of Surface 151

1. Concept of Surface 151

2. Regular Surfaces 152

3. Tangent Plane to a Surface 153

4. Equation of a Tangent Plane 155

5. Osculating Paraboloid of a Surface 156

6. Classification of Surface Points 158

Exercises to Chapter X 159

Chapter XI. Surface Curvature 161

1. Surface Linear Element 161

2. Area of a Surface 162

3. Normal Curvature of a Surface 164

4. Indicatrix of the Normal Curvature 165

5. Conjugate Coordinate Lines on a Surface 167

6. Lines of Curvature 168

7. Mean and Gaussian Curvature of a Surface 170

8. Example of a Surface of Constant Negative Gaussian Curvature 172

Exercises to Chapter XI 173

Chapter XII. Intrinsic Geometry of Surface 175

1. Gaussian Curvature as an Object of the Intrinsic Geometry of Surfaces 175

2. Geodesic Lines on a Surface 178

3. Extremal Property of Geodesics 179

4. Surfaces of Constant Gaussian Curvature 180

5. Gauss-Bonnet Theorem 181

6. Closed Surfaces 182

Exercises to Chapter XII 184

Part Three. Foundations of Geometry 186

Chapter XIII. Historical Survey 186

1. Euclid's Elements 186

2. Attempts to Prove the Fifth Postulate 188

3. Discovery of Non-Euclidean Geometry 189

4. Works on the Foundations of Geometry in the Second Half of the 19th century 191

5. System of Axioms for Euclidean Geometry according to D. Hilbert 192

Chapter XIV. System of Axioms for Euclidean Geometry and Their Immediate Corollaries 194

1. Basic Concepts 194

2. Axioms of Incidence 195

3. Axioms of Order 196

4. Axioms of Measure for Line Segments and Angles 197 5. Axiom of Existence of a Triangle Congruent to a Given One 199

6. Axiom of Existence of a Line Segment of Given Length 200

7. Parallel Axiom 202

8. Axioms for Space 202

Chapter XV. Investigation of Euclidean Geometry Axioms 203

1. Preliminaries 203

2. Cartesian Model of Euclidean Geometry 204

3. "Betweenness" Relation for Points in a Straight Line. Verification of the Axioms of Order 205

4. Length of a Segment. Verification of the Axiom of Measure for Line Segments 207

5. Measure of Angles in Degrees. Verification of Axiom III* 208

6. Validity of the Other Axioms in the Cartesian Model 210

7. Consistency and Completeness of the Euclidean Geometry Axiom System 212

8. Independence of the Axiom of Existence of a Line Segment of Given Length 214

9. Independence of the Parallel Axiom 216

10. Lobachevskian Geometry 218

Chapter XVI. Projective Geometry 222

1. Axioms of Incidence for Projective Geometry 222

2. Desargues Theorem 223

3. Completion of Euclidean Space with the Elements at Infinity 225

4. Topological Structure of a Projective Straight Line and Plane 226

5. Projective Coordinates and Projective Transformations 228

6. Cross Ratio 230

7. Harmonic Separation of Pairs of Points 232

8. Curves of the Second Degree and Quadric Surfaces 233

9. Steiner Theorem 235

10. Pascal Theorem 236

11. Pole and Polar 238

12. Polar Reciprocation. Brianchon Theorem 240

13. Duality Principle 241

14. Various Geometries in Projective Outlook 243

Exercises to Chapter XVI 245

Part Four. Certain Problems of Elementary Geometry 247

Chapter XVII. Methods for Solution of Construction Problems 247

1. Preliminaries 247

2. Locus Method 248

3. Similarity Method 250

4. Reflection Method 251

5. Translation Method 251

6. Rotation Method 252

7. Inversion Method 253

8. On Solvability of Construction Problems 255

Exercises to Chapter XVII 256

Chapter XVIII. Measuring Lengths, Areas and Volumes 258

1. Measuring Line Segments 258

2. Length of a Circumference 260

3. Areas of Figures 261

4. Volumes of Solids 265

5. Area of a Surface 267

Chapter XIX. Elements of Projection Drawing 268

1. Representation of a Point on an Epure 268

2. Problems Leading to a Straight Line 269

3. Determination of the Length of a Line Segment 270

4. Problems Leading to a Straight Line and a Plane 271

5. Representation of a Prism and a Pyramid 273

6. Representation of a Cylinder, a Cone and a Sphere 274

7. Construction of Sections 275

Exercises to Chapter XIX 277

Chapter XX. Polyhedral Angles and Polyhedra 278

1. Cosine Law for a Trihedral Angle 278

2. Trihedral Angle Conjugate to a Given One 279

3. Sine Law for a Trihedral Angle 280

4. Relation Between the Face Angles of a Polyhedra Angles 281

5. Area of a Spherical Polygon 282

6. Convex Polyhedra. Concept of Convex Body 283

7. Euler Theorem for Convex Polyhedra 284

8. Cauchy Theorem 285

9. Regular Polyhedra 288

Exercises to Chapter XX 289

Answers to Exercises, Hints and Solutions 291

IdentifierPogorelovGeometryMir1987

Identifier-arkark:/13960/t23c3b949

OcrABBYY FineReader 11.0 (Extended OCR)

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Year1987

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